The concept of brief encounters, even romantic encounters, with a stranger recurs often in the verses of Walt Whitman.

Take, for example, these lines from one of the inscriptions that Whitman wrote to his 1860 edition of Leaves of Grass.
"Stranger, if you passing meet me and desire to speak to me,
why should you not speak to me?
And why should I not speak to you?"

Clearly, Walt Whitman sees brief, chance encounters with strangers as an appropriate opportunity for the strangers to interact. Perhaps the communication will allow the strangers to become friends.

In the lines of "To A Stranger," Whitman indicates that the strangers might become intimate and affectionate friends. The narrator in the poem is comfortably able to imagine himself creating a past history with the passing stranger and to foresee the opportunities for them to enjoy each other in physically affectionate ways.

Here